Notes for Intel® oneAPI Math Kernel Library Vector Statistics

Developer Reference

Contents

ARS5

This is a keyed family of counter-based BRNGs. The state consists of 128-bit integer counter 
c
 and a 128-bit key
k
. The BRNG is based on the AES encryption algorithm [FIPS-197]. The 32-bit output is obtained in the following way [Salmon2011]:
  1. The
    i
    -th number is defined by the following formula:
    r
    i
     =
    (
    f(i/4
    ) >> ((i mod 4) * 32) mod 2
    32
  2. Function
    f(c
    )
     takes 128-bit input and produces 128-bit result obtained in the following way:
    1. Put
      c
      0
       =
      c xor k
       and
      k
      0
       =
      k
      .
    2. The following recurrence is calculated N times:
      1. c
        i+1
         =
        SubBytes(
        c
        )
      2. c
        i+1
         =
        ShiftRows(
        c
        i+1
        )
      3. c
        i+1
         =
        MixColumns(
        c
        i+1
        )
        , this step is omitted if
        i + 1 = N
      4. c
        i+1
         =
        AddRoundKey(
        c
        i+1
        ,
        k
        j
        )
      5. Lo(k
        i+1
        )
         =
        Lo(
        k
        )
         + 0x9E3779B97F4A7C15
        Hi(k
        i+1
        )
         =
        Hi(
        k)
         + 0xBB67AE8584CAA73B
    3. Put
      f(
      c
      ) = cN
      , where
      N
       = 5
  3. Real output:
    u
    n
    = (int)r
    n
    /2
    32
    + ½
Specification for the
SubBytes
,
ShiftRows
,
MixColumns
 and
AddRoundKey
 functions can be found in [FIPS-197].

Real Implementation (Single and Double Precision)

The output vector is the sequence of the floating-point values
u
0
,
u
1
, ...

Integer Implementation

The output vector of 32-bit integers
r
0
,
r
1
, ...

Stream Initialization by Function
vslNewStream

ARS5 generates the stream and initializes it specifying the 32-bit input integer parameter seed. The stream state is two 128-bit numbers
c
 and
k
 initialized in the following way:
  1. Assume
    k
     = seed.
  2. Assume
    c
     = 0.

Stream Initialization by Function
vslNewStreamEx

ARS5 generates the stream and initializes it specifying the array
params[]
 of
n
 32-bit integers:
  1. If
    n
     = 0, assume
    c
     =
    k
     = 0.
  2. If
    n
     = 1, assume
    k
     = params[0],
    c
     = 0.
  3. If
    n
     = 2, assume
    k
     = params[0] + params[1]*2
    32
    ,
    c
     = 0.
  4. If
    n
     = 3, assume
    k
     = params[0] + params[1]*2
    32
     + params[2]*2
    64
    ,
    c
     = 0.
  5. If
    n
     = 4, assume
    k
     = params[0] + params[1]*2
    32
     + params[2]*2
    64
     + params[3]*2
    96
    ,
    c
     = 0.
  6. If
    n
     = 5, assume
    k
     = params[0] + params[1]*2
    32
     + params[2]*2
    64
     + params[3]*2
    96
    ,
    c
     = params[4].
  7. If
    n
     = 6, assume
    k
     = params[0] + params[1]*2
    32
     + params[2]*2
    64
     + params[3]*2
    96
    ,
    c
     = params[4] + params[5]*2
    32
    .
  8. If
    n
     = 7, assume
    k
     = params[0] + params[1]*2
    32
     + params[2]*2
    64
     + params[3]*2
    96
    ,
    c
     = params[4] + params[5]*2
    32
     + params[6]*2
    64
    .
  9. If
    n
     >= 8, assume
    k
     = params[0] + params[1]*2
    32
     + params[2]*2
    64
     + params[3]*2
    96
    ,
    c
     = params[4] + params[5]*2
    32
     + params[6]*2
    64
     + params[7]*2
    96
    .

Subsequences Selection Methods

vslSkipAheadStream
Supported 
vslSkipAheadStreamEx
Supported 
vslLeapfrogStream
Not supported

Generator Period

Empirical Testing Results Summary

Test Name 
vsRngUniform
vdRngUniform
viRngUniform
viRngUniformBits
3D Spheres Test
OK (20% errors)
OK (20% errors)
Not applicable
OK (20% errors)
Birthday Spacing Test
Not applicable
Not applicable
Not applicable
OK (0% errors)
Bitstream Test
Not applicable
Not applicable
Not applicable
OK (15% errors)
Rank of 31x31 Binary Matrices Test
Not applicable
Not applicable
Not applicable
OK (0% errors)
Rank of 32x32 Binary Matrices Test
Not applicable
Not applicable
Not applicable
OK (0% errors)
Rank of 6x8 Binary Matrices Test
Not applicable
Not applicable
Not applicable
OK (0% errors)
Counts-the-1’s Test (stream of bits)
Not applicable
Not applicable
Not applicable
OK (0% errors)
Counts-the-1’s Test (stream of specific bytes)
Not applicable
Not applicable
Not applicable
OK (0% errors)
Craps Test
OK (30% errors)
OK (30% errors)
OK (30% errors)
OK (30% errors)
Parking Lot Test
OK (10% errors)
OK (10% errors)
Not applicable
OK (10% errors)
2D Self-Avoiding Random Walk Test
OK (20% errors)
OK (10% errors)
Not applicable
OK (10% errors)
  1. The tabulated data is obtained using the one-level (threshold) testing technique. The OK result indicates FAIL < 50%. The run fails when p-value falls outside the interval [0.05, 0.95].
  2. The stream tested is generated by calling the function
    vslNewStream
     with seed=7,777,777.

Product and Performance Information

1

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.